Neyman-Pearson lemma

For testing one simple hypothesis against another, a test that rejects for sufficiently large likelihood ratio has greatest power among all tests of no greater size.

For independent Weibull distribution observations of known shape two, the likelihood ratio between any scale ${\lambda }_1>{\lambda }_0$ and the null scale ${\lambda }_0$ increases with ${\sum }_i{X}_i^2$. Under the null, $2{\sum }_i{X}_i^2/{\lambda }_0^2$ has a chi-squared distribution with $2n$ degrees of freedom. The same upper-tail critical region is therefore most powerful for every larger scale, giving a uniformly most powerful size-$\alpha$ test.

For independent $N(0,{\sigma }^2)$ observations, testing a fixed variance against larger variances uses the sum of squares $S={\sum }_i{X}_i^2$. The likelihood ratio for any simple larger variance is strictly increasing in $S$. The Neyman-Pearson lemma therefore gives the displayed rejection rule, where $F_n$ is the chi-squared distribution function with $n$ degrees of freedom. Its threshold is independent of the selected larger variance, so it is a uniformly most powerful test for the one-sided composite alternative. Its power function is $1-F_n(({\sigma }_0^2/{\sigma }^2)F_n^{-1}(1-\alpha ))$, increasing continuously from zero to one and equaling $\alpha$ at the null. A known nonzero mean is handled by replacing each observation by its difference from that mean.

For an independent statistical sample from a gamma distribution with known shape $\lambda >0$ and unknown rate $\theta >0$, test $\theta ={\theta }_0$ against $\theta >{\theta }_0$. The alternative-to-null likelihood ratio for any larger rate ${\theta }_1$ is $({\theta }_1/{\theta }_0{)}^{n\lambda }\exp \{-({\theta }_1-{\theta }_0)S\}$, where $S={\sum }_i{X}_i$. This decreases in $S$, so the Neyman-Pearson lemma gives one common lower-tail ordering for all alternatives.

Let $G_a$ be the unit-rate gamma distribution cumulative distribution function with shape $a$. The uniformly most powerful test of exact test size $\alpha \in (0,1)$ rejects when $S<c_{\alpha }$, where $c_{\alpha }=G_{n\lambda }^{-1}(\alpha )/{\theta }_0$. Its power function is displayed above. If $n\lambda =1$, the sum has an exponential distribution, so $c_{\alpha }=-\log (1-\alpha )/{\theta }_0$ and $\pi (\theta )=1-(1-\alpha {)}^{\theta /{\theta }_0}$. Monotonicity of $G_a(\theta c_{\alpha })$ also makes this test valid and uniformly most powerful for the larger null hypothesis $\theta \le {\theta }_0$.

For two unit-scale Laplace distributions with locations ${\theta }_0<{\theta }_1$, the likelihood ratio is nondecreasing in $X$, with constant tails and an increasing middle interval. Therefore an upper-tail statistical test of exact size of a statistical test $\alpha$ is a most powerful test. Its cutoff is $c_{\alpha }={\theta }_0-\log (2\alpha )$ for $\alpha \le 1/2$, and $c_{\alpha }={\theta }_0+\log (2(1-\alpha ))$ otherwise. Under location ${\theta }_1$, its statistical power is $\frac{1}{2}{e}^{{\theta }_1-c_{\alpha }}$ if $c_{\alpha }\ge {\theta }_1$, and $1-\frac{1}{2}{e}^{c_{\alpha }-{\theta }_1}$ otherwise. Constant likelihood ratio tails can make the most powerful test nonunique.

For two simple hypotheses with probability densities $f_0,f_1$, a critical region $C$ has sum of Type I error and Type II error probabilities ${\int }_C{f}_0+{\int }_{C^c}{f}_1$. Pointwise minimization chooses $C=\{f_1>f_0\}$, with either decision allowed on the equality set. Thus the optimal likelihood ratio threshold is $1$, and the minimum is $\int \min (f_0,f_1)=1-\frac{1}{2}\int |f_1-f_0|$. This criterion weights the two error probabilities equally; unequal costs or prior probabilities change the threshold.

For independent exponential distribution observations with rate $\lambda$, the likelihood ratio for a larger simple rate is decreasing in their sum. The Neyman-Pearson lemma therefore selects a lower-tail sum threshold $c$. The sum has a gamma distribution, giving the displayed power; its derivative is $c{e}^{-\lambda c}(\lambda c{)}^{n-1}/(n-1)!>0$. Calibrating at ${\lambda }_0$ gives size $\alpha$ for the composite null $\lambda \le {\lambda }_0$ and a uniformly most powerful test against larger rates.

For independent samples from an exponential distribution, a test of rate at most ${\lambda }_0$ against rate greater than ${\lambda }_0$ rejects for small sample sums. All simple-alternative likelihood ratios against ${\lambda }_0$ decrease with that sum; its gamma distribution calibrates the size at ${\lambda }_0$. Monotonicity of the rejection probability in the rate proves that this is a uniformly most powerful test for the composite hypotheses.

A one-parameter family has a monotone likelihood ratio in a statistic $T$ when $f(x\mid {\theta }_1)/f(x\mid {\theta }_0)$ is nondecreasing in $T(x)$ whenever ${\theta }_1>{\theta }_0$.

For a family with a monotone likelihood ratio in $T$, a prior density $\pi$ and cutoff $a$, the posterior odds of $\theta >a$ against $\theta \le a$ are nondecreasing in $T$. To prove it, write the unnormalized masses as $A(t)$ and $B(t)$. For $t_2>t_1$, the difference $A(t_2)B(t_1)-A(t_1)B(t_2)$ is the double integral over $\theta >a\ge \eta$ of $\pi (\theta )\pi (\eta )[f_{\theta }(t_2)f_{\eta }(t_1)-f_{\theta }(t_1)f_{\eta }(t_2)]$. Each integrand is nonnegative by the likelihood-ratio ordering. Common factors in the full-data likelihood cancel from the odds, so the same argument works when $T$ is a statistic rather than the complete observation. Interpret zero denominators by extended odds whenever the posterior is defined.

An upper-tail monotone test in a real statistic $T$ has rejection probability $\varphi =1_{T>c}+\eta 1_{T=c}$, with $0\le \eta \le 1$; the constant tests correspond to infinite cutoffs. For a family with positive densities on a common support and a monotone likelihood ratio in $T$, its power function can cross that of an arbitrary competing test only from below to above as the parameter increases. If $r=f_{{\theta }_1}/f_{{\theta }_0}$ and ${\theta }_1>{\theta }_0$, choose a positive number $k$ between the ratios on opposite sides of the cutoff. The signs of $\varphi -{\varphi }^{\prime }$ and $r-k$ agree away from $T=c$, and $r=k$ at a boundary with positive mass. Therefore $E_{{\theta }_1}(\varphi -{\varphi }^{\prime })\ge k{E}_{{\theta }_0}(\varphi -{\varphi }^{\prime })$. In particular strict positivity at ${\theta }_0$ propagates to ${\theta }_1$. Positivity of the densities matters for this strict conclusion; with changing supports only a weak conclusion may remain.

For a family with a monotone likelihood ratio in $T$, a level-$\alpha$ upper-tail test in $T$ is a uniformly most powerful test for the corresponding one-sided hypothesis.

For one observation from the logistic distribution, the size-$\alpha$ test of $H_0:\theta \le 0$ against $H_1:\theta >0$ rejects when